Question

A square and an equilateral triangle have equal perimeters. The area of the triangle is $2\sqrt {3}$ square inches. What is the number of inches in the length of the diagonal of the square?

Answer 1

An equilateral triangle with side length $x$ has area $\dfrac{\sqrt3}4x^2$, so that

$\dfrac{\sqrt3}4x^2=2\sqrt3\implies x^2=8\implies x=2\sqrt2$

Then the triangle, and hence the square, has a perimeter of $3x=6\sqrt2$.

The perimeter of a square with side length $y$ is $4y$, so that

$4y=6\sqrt2\implies y=\dfrac{3\sqrt2}2$

The length of the diagonal of any square is $\sqrt2$ longer than the length of its side, so that this square's diagonal length is

$\sqrt2\,y=\dfrac{3(\sqrt2)^2}2=\boxed{3}$